CN113156365B - Joint Estimation Method of Velocity, Angle and Distance Based on Conjugate ZC Sequence Pairs - Google Patents

Abstract

The invention belongs to the technical field of wireless positioning, in particular to a method for joint estimation of velocity, angle and distance based on conjugate ZC sequence pairs. The invention includes: at the transmitting end, designing a pair of conjugated ZC sequences as the transmitting sequence; at the receiving end, receiving multi-path signals with different transmission delays, Doppler frequency offsets and angles, and using the maximum likelihood method to carry out Parameter estimation; using the alternate projection method to transform the high-dimensional parameter estimation problem into multiple low-dimensional parameter estimation problems; for single-path parameter estimation, based on the properties of the ZC sequence, the two-dimensional estimation of time delay and frequency offset is converted into two One-dimensional estimation, then combined with Newton iteration for precise estimation. The invention can realize high-precision transmission delay, Doppler frequency offset and angle estimation under the condition of low bandwidth. Simulation shows that the present invention can realize the distance estimation accuracy of 1cm, the speed estimation accuracy of 1m/s and the angle estimation accuracy of 0.01° in the case of 20MHz bandwidth.

Description

Joint Estimation Method of Velocity, Angle and Distance Based on Conjugate ZC Sequence Pairs

technical field

The invention belongs to the technical field of wireless positioning, and in particular relates to a method for joint estimation of velocity, angle and distance based on conjugate ZC sequence pairs.

Background technique

Wireless ranging and positioning algorithms have practical applications in life, and have broad applications in indoor positioning, Internet of Vehicles, automatic driving and other fields. With the development of 5G and IoT technologies, technologies related to positioning are attracting more and more attention. Market research firm Markets and Markets predicted in an analysis report published in 2020 that the global indoor positioning market size will rise from $7.11 billion in 2017 to $40.99 billion in 2022 [1]. However, in some scenarios where the Global Positioning System (GPS) cannot cover or in the future 5G autonomous driving, technologies that can achieve high-resolution positioning are urgently needed, and even centimeter-level positioning accuracy is required.

Most of the existing research work only considers the joint estimation of delay and angle, or cannot distinguish multipath dense signals well, or does not consider the non-integer Nyquist sampling delay. However, in practice, due to the relative motion of the transceiver, there will be a Doppler frequency offset. By estimating the Doppler frequency offset, that is, the speed, a better positioning can be performed. At present, in terms of ranging accuracy, Ultra Wide Band (UWB) ) technology can achieve centimeter-level accuracy, but this technology requires large bandwidth and high hardware requirements. In the case of limited bandwidth, the estimation of non-integer Nyquist sampling delay can improve the accuracy of ranging. And reduce the cost of hardware; considering the time delay, Doppler frequency offset, and angle of each path, synthesizing these information is beneficial to environmental perception and positioning. Therefore, how to realize the super-resolution estimation of time delay, Doppler frequency offset and angle in multipath environment is an urgent problem to be solved.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a method for joint estimation of distance (transmission delay), angle and velocity (Doppler frequency offset) in a wireless multipath environment with low computational complexity and low hardware implementation complexity.

In the method for joint estimation of distance, angle and velocity in wireless multipath environment provided by the present invention, a pair of conjugated Zadoff-Chu sequences (ZC sequences) are designed at the transmitting end as the transmitting sequence; Transmission delay, Doppler frequency offset, angle multipath signal, use maximum likelihood method to estimate parameters, and then use alternate projection method to convert the original high-dimensional parameter estimation problem (multipath) into multiple low-dimensional parameters Parameter estimation problem (single-path), avoiding high-dimensional search; for single-path parameter estimation, based on the properties of ZC sequences, the two-dimensional estimates of delay and frequency offset are converted into two one-dimensional estimates, which further reduces the computational cost complexity, and then combined with Newton iteration to estimate the exact value.

Specific steps are as follows

The first step is to design a pair of conjugated ZC sequences as the transmission sequence;

In the second step, based on the properties of the conjugated ZC sequence pair, the initial solutions of time delay, angle, and Doppler frequency offset are simultaneously obtained;

In the third step, based on the initial solutions of time delay, angle, and Doppler frequency offset, the precise values of the three-dimensional parameters are further obtained through Newton iteration.

Among them, assuming that there is a uniform linear array with M antennas at the receiving end, the array response of the signal to the array with respect to the angle θ can be expressed as:

Among them, d represents the spacing of the antenna array, λ represents the wavelength, j represents the imaginary number, and (·) ^T represents the transposition operation.

Considering the multipath environment, the multipath signal received by the receiving end includes a direct path and U-1 reflection paths, and each path has different delays, angles, and Doppler frequency offsets. The multipath signal model is as follows:

表示实数域。Among them, β _u , θ _u , τ _u , ξ _u represent the channel gain, angle of arrival, time delay and Doppler frequency offset of the u-th path signal; y(t) represents the received signal, and x(t) represents the training sequence , z(t) represents Gaussian noise obeying complex Gaussian distribution, U is the number of multipath signals,

represents the real number field.

In the first step, a pair of conjugated Zadoff-Chu sequences are designed, and the specific process is as follows:

的ZC序列[2]：(1) For a length of

The ZC sequence [2]:

为正整数，r是和

互质的正整数参数。可以看出

即ZC序列是周期的，因此我们可以改变ZC序列的索引范围：对于

为偶数，索引范围改为

而对于

为奇数，索引范围为

假设

是偶数，那么对于一个整数时延τ则有：in,

is a positive integer, and r is the sum

Coprime positive integer argument. As can be seen

That is, the ZC sequence is periodic, so we can change the index range of the ZC sequence: for

is even, the index range is changed to

And for

is odd, the index range is

Assumption

is even, then for an integer delay τ there are:

的频偏；对于

长度是奇数的情况，ZC序列的这种时延-频偏互相转换的性质依然成立。不失一般性，我们令r＝1，因此后面可以省略r。This shows that for a ZC sequence, an integer delay τ corresponds to

frequency offset; for

When the length is an odd number, the time delay-frequency offset mutual conversion property of the ZC sequence still holds. Without loss of generality, we set r = 1, so r can be omitted later.

(2) The time delay-frequency offset interchange property of the ZC sequence described in formula (4) is only applicable to integer time delay. In order to perform super-resolution time delay estimation, the influence of the shaping filter existing at the transceiver end needs to be considered. Assuming that the shaping filter at the transceiver end is a raised cosine filter, the impulse response of the raised cosine filter can be expressed as:

where α is the roll-off coefficient, and T _s is the Nyquist sampling period; the discrete ZC sequence s(n) can be expressed as a continuous-time signal x(t) after passing through the shaping filter:

的ZC序列经过升余弦滤波器后的模值，可以看到中间的低频部分几乎不受影响，两端的高频部分会受到明显压制。Due to the low-pass nature of the shaping filter, the high frequency part of the ZC sequence is suppressed. Figure 1 is a

The modulus value of the ZC sequence after passing through the raised cosine filter, it can be seen that the low frequency part in the middle is hardly affected, and the high frequency part at both ends will be significantly suppressed.

Further, the low-frequency part of length L in the middle of the demonstration continuous-time signal x(t) is approximated as a chirp signal, namely:

从图1也可以看出，滚降系数α也会影响L的选取。图2展示了在α＝0.3时，

的波形，其中，|·|表示取模操作。可以看出对于L＝250，公式(7)的近似是合适的。where L is a positive integer and

It can also be seen from Figure 1 that the roll-off coefficient α also affects the selection of L. Figure 2 shows that when α=0.3,

, where |·| represents the modulo operation. It can be seen that for L=250, the approximation of equation (7) is suitable.

Therefore, the signal after the low frequency part of the ZC sequence passes through the raised cosine filter can be regarded as a chirp signal, and there is also an exchange relationship between time delay and frequency offset, namely:

The time delay τ can be arbitrary, and is no longer limited to an integer multiple of the Nyquist sampling period.

(3) Based on the approximation of formula (7), consider a pair of conjugated ZC sequences, denoted as s(n) and s ^* (n), respectively:

为偶数，

(·)^*表示取共轭操作。对于前一半ZC序列s(n)，发送时为其加上一个长为

的前缀和长为

的后缀：in,

is an even number,

(·) ^* indicates the conjugation operation. For the first half of the ZC sequence s(n), add a length of

The prefix and length of

suffix:

where Q is a positive integer. Similarly, the corresponding prefix and suffix are also added to the latter half of the ZC sequence. The format of the sent training sequence is shown in Figure 3. The prefix and suffix are designed to resist inter-symbol interference (ISI) and frequency offset effects. When the receiving end is processing, part of the prefix and suffix should be removed from the received signal. Taking the first half of the training sequence as an example, as shown in Figure 4, the received signal with a length of L+Q will be affected by ISI interference and frequency offset with a length of R ₁ (equivalent to a time offset with a length of R ₂ ) Therefore, after receiving the signal, the receiving end performs the operation of removing the prefix and suffix, and selects the beginning of the intercepted sequence from the area S, thereby ensuring that the intercepted L-length sequence is not affected by ISI interference and frequency offset.

In the second step, the initial solutions of the time delay, angle, and Doppler frequency offset are obtained, and the specific process is as follows:

(1) Based on the model formula (2), considering the first half of the ZC sequence, the sampling of the received signal can be expressed as:

是任意大于零的实数，T_s是奈奎斯特采样周期。不失一般性，令T_s＝1，得到：in,

is any real number greater than zero, and T _s is the Nyquist sampling period. Without loss of generality, let T _s =1, we get:

Rewrite (14) into the following form:

Y=A(θ)diag(β)X(τ,ξ) ^T +Z (15)

in,

X(τ,ξ)=[x(τ ₁ )⊙d(ξ ₁ ), x(τ ₂ )⊙d(ξ ₂ ),...,x(τ _U )⊙d(ξ _U )] (19 )

代表复数域，diag(·)表示向量对角化为矩阵,⊙表示哈达玛积。in,

Represents the field of complex numbers, diag( ) represents the diagonalization of a vector into a matrix, and ⊙ represents the Hadamard product.

Since the noise follows a complex Gaussian distribution, the maximum likelihood estimate of the parameters {β, τ, ξ, θ} can be written in the least squares form:

所以有：where τ=[τ ₁ , τ ₂ ,...,τ _U ] ^T ,ξ=[ξ ₁ ,ξ ₂ ,...,ξ _U ] ^T ,θ=[θ ₁ ,θ ₂ ,... , θ _U ] ^T , || · || _F denotes the F norm. because

F:

表示克罗内克积，Among them, vec( ) represents the matrix column vectorization,

represents the Kronecker product,

So (22) can be rewritten as:

in,

Then the maximum likelihood estimate of β can be expressed as:

where (·) ^H represents the transpose conjugation operation. Substituting (28) into (26), we get:

in,

表示投影矩阵。in,

represents the projection matrix.

和时延

后，可以根据速度

其中c是光速，f_c是载频，从而估计出发送端相对接收端的速度，根据距离

可以估计出距离。Doppler frequency offset is estimated

and delay

After that, according to the speed

Where c is the speed of light, f _c is the carrier frequency, so as to estimate the speed of the sender relative to the receiver, according to the distance

Distance can be estimated.

(2) Considering the conjugated ZC sequence pair, then:

in,

对应前一半ZC序列，x₂(τ_u，ξ_u)和

对应后一半ZC序列。Here, x ₁ (τ _u , ξ _u ) and

Corresponding to the first half of the ZC sequence, x ₂ (τ _u , ξ _u ) and

Corresponding to the latter half of the ZC sequence.

make:

是将

从

中删去后得到的结果。利用投影矩阵的性质，得到：and

will

from

The result obtained after removing the . Using the properties of the projection matrix, we get:

表示矩阵

的正交投影矩阵。in,

representation matrix

The orthographic projection matrix of .

Substituting equation (37) into equation (29), we get:

已知，即给定

则公式(38)可以化简为：Assume that the multipath numbers U and

known, given

Then formula (38) can be simplified as:

Therefore, the multi-path problem is transformed into multiple single-path problems by the method of alternate projection, avoiding high-dimensional search.

(3) For the solution of the u-th path problem, first estimate the initial values of time delay, angle and frequency offset. The initial values of the parameters of the u-th path are achieved using the following approximations:

in,

则有：because

Then there are:

和

分别是

和

按列重构的矩阵。将(41)(42)代入(40)，得到：in,

and

respectively

and

Column-wise reconstructed matrix. Substituting (41)(42) into (40), we get:

Substitute (32), (33) into (43) to get:

make:

Then the estimates of τ _u , ξ _u , θ _u can be transformed into estimates of η _u , ζ _u , θ _u :

Considering the first half of the ZC sequence first, suboptimal solutions for ζ _u and θ _u can be obtained:

做二维快速傅里叶变换，然后找出使模值最大的坐标，从而求出

through the pair

Do a two-dimensional fast Fourier transform, and then find the coordinate that maximizes the modulus value, so as to find

已知，则有：At this time, consider the second half of the ZC sequence, at this time

known, there are:

做一维快速傅里叶变换，找出使模值最大的坐标，从而得到

Right now

Do a one-dimensional fast Fourier transform to find the coordinates that maximize the modulus value, and get

的初始值，进而可以得到时延和多普勒频偏的初始值：Combining the estimation results of (47) (48), we can get

The initial value of , and then the initial value of delay and Doppler frequency offset can be obtained:

In the third step, the exact value of the three-dimensional parameter is further obtained through the Newton iteration, and the specific process is as follows:

本发明利用牛顿迭代[4]得到精确值。令Λ表示目标函数(39)，即：Based on the obtained initial value

The present invention utilizes Newton iteration [4] to obtain the exact value. Let Λ denote the objective function (39), namely:

where ψ=[τ _u , ξ _u , θ _u ] ^T . The iterative formula of Newton iteration is:

ψ ⁽ⁱ⁺¹⁾ = ψ ⁽ⁱ⁾ -sH ^-1 g (51)

和

分别表示关于目标函数Λ的黑塞矩阵和雅各比向量，s表示步长，并且由回溯直线法[4]优化得到。in,

and

respectively represent the Hessian matrix and Jacobian vector about the objective function Λ, s represents the step size, and is optimized by the backtracking line method [4].

The algorithm flow of the alternate projection method is as follows:

When initializing the alternate projection process, assuming that the received signal is single-path, combined with formula (29), the three-dimensional parameters can be obtained by using the following formula:

是常数因此可以省略。从而估计出

其中上标代表迭代次数，因此也就得到了

令

利用公式(39)估计出

接着，令

从而估计

这个过程一直进行，到

估计出

in,

is a constant and can therefore be omitted. thus estimating

where the superscript represents the number of iterations, so we get

make

Using Equation (39) to estimate

Next, let

thus estimating

This process continues until

estimated

根据公式(39)估计出

然后利用

估计出

以此类推到利用

估计出

重复上述过程直到迭代收敛。In the second iteration, first use

According to formula (39), it is estimated that

then use

estimated

And so on to use

estimated

The above process is repeated until the iterations converge.

The advantages of the method of the present invention:

(1) The present invention can estimate the time delay, Doppler frequency offset and angle of each path from the multipath signal, thereby realizing ranging, speed and direction finding.

(2) The present invention considers the influence of the shaping filter in real hardware, and can perform super-resolution time delay estimation.

(3) The present invention avoids high-dimensional search through alternate iteration, and has low hardware implementation complexity.

(4) The present invention designs a transmission sequence based on the ZC sequence, which can convert the two-dimensional search on time delay and frequency offset into two one-dimensional searches, thereby reducing the computational complexity.

(5) The method used in the present invention is superior to the existing SAGE [3] method in estimation accuracy.

The present invention can realize high-precision transmission delay, Doppler frequency offset and angle estimation under the condition of low bandwidth, thereby estimating the moving speed of the sending end and the distance and angle from the sending end to the receiving end. High-precision positioning. Simulation shows that the present invention can achieve a distance estimation accuracy of 1cm, a speed estimation accuracy of 1m/ and an angle estimation accuracy of 0.01° in the case of a 20MHz bandwidth.

Description of drawings

Figure 1 is a schematic diagram of the angle of arrival of a signal with respect to a linear uniform array in a narrowband far-field environment.

Figure 2 is a modulus diagram of the ZC sequence after shaping the filter.

Figure 3 is a graph showing that the approximation of formula (7) is good (|x(t)-s(t)|<1.8x10-3 for -125≤t<125).

FIG. 4 is a schematic diagram of the format of the transmitted training sequence based on the conjugated ZC sequence.

Figure 5 is a schematic diagram of prefix and suffix removal.

Figure 6 is a graph of the estimated RMSE performance of velocity (a), angle (b), and distance (c) in the case of two diameters.

Figure 7 is the contour map of the direct diameter estimation performance under different delay and angle differences in the case of two diameters (the first three pictures: (a), (b), (c)) and the corresponding Cramerau line ( CRB) contour plots (last 3 plots: (d), (e), (f)).

FIG. 8 is a performance comparison diagram (frequency offset) between the method of the present invention and the method of SAGE.

Figure 9 is a performance comparison diagram (angle) between the method of the present invention and the method of SAGE.

FIG. 10 is a performance comparison diagram (time delay) between the method of the present invention and the method of SAGE.

Detailed ways

The present invention is further described below through specific embodiments.

每一个ZC序列各自都有一个长

的前缀和长为

的后缀。成型滤波器采用升余弦滤波器，滚降系数α＝0.3。假设接收端有6个天线，天线间间距等于半波长即

载波频率f_c＝2.4GHz，带宽B＝20MHz，也即意味着T_s＝50ns。每个仿真结果都进行了1000次蒙特卡洛。As an embodiment, the present invention simulates the complete process of signal transmission-shaping filter-passing channel-receiving signal-signal processing with a computer. The conjugated ZC sequence pair to be sent is L=250,

Each ZC sequence has a long

The prefix and length of

suffix. The shaping filter adopts a raised cosine filter, and the roll-off coefficient α=0.3. Assuming that there are 6 antennas at the receiving end, the spacing between the antennas is equal to half a wavelength, i.e.

The carrier frequency f _c =2.4GHz, and the bandwidth B = 20MHz, which means T _s =50ns. Each simulation result was performed 1000 times of Monte Carlo.

其中φ₁和φ₂是随机产生的随机数。两径的传输距离和速度分别对应ρ＝[18，19.5]m和v＝[75，175]m/s。图6为这两径信号的速度、角度、距离估计的根均方误差(RMSE)曲线以及它们的克拉美劳线(CRB)。仿真表明本发明可以有效估计出多径信号的速度、角度、距离，并且可以达到1m/s的测速精度，1cm的测距精度，0.01°的测角度精度。Embodiment 1, consider that there are two paths in total, the frequency offset is ξ=[3×10 ^-5 , 7×10 ^-5 ]T _s , the angle is θ=[5°, 20°], and the time delay is τ=[ 1.2, 1.3] Ts, channel gain

where _φ1 and _φ2 are randomly generated random numbers. The transmission distance and speed of the two paths correspond to ρ=[18, 19.5]m and v=[75,175]m/s, respectively. Figure 6 shows the root mean square error (RMSE) curves of velocity, angle, and distance estimates for the two path signals and their Cramerau lines (CRB). Simulation shows that the present invention can effectively estimate the speed, angle and distance of multipath signals, and can reach the speed measurement accuracy of 1m/s, the distance measurement accuracy of 1cm, and the angle measurement accuracy of 0.01°.

In Embodiment 2, considering the situation of two paths with the same channel gain, the situation where the estimation result changes with the change of the angle difference and the time delay difference between the two paths is explored. The frequency offset of the two paths is ξ=[10 ^-5 , 10 ^-5 ]/T _s , the angle is θ=[10°, 10°+Δθ], the time delay is τ=[1.1, 1.1+Δτ]T _s , the signal The noise ratio is 20dB. The first three pictures in Figure 7 show the RMSE results of the frequency offset, angle, and time delay of the first path signal, and the last three pictures show the corresponding CRB. The simulated contours show that the present invention can distinguish multipaths by utilizing the differences in time and space even in a harsh environment with dense multipaths, and the simulation performance can approach CRB.

Embodiment 3, considering the two-path situation of the same channel gain, compare the estimation accuracy between the alternate projection method used in the present invention and the SAGE method. The frequency offset of the two paths is ξ=[10 ^-5 , 10 ^-4 ]/T _s , the angle is θ=[10°, 15°], the time delay is τ=[1.1, 1.1+Δτ]T _s , the signal-to-noise ratio is 20dB. Figure 8, Figure 9, and Figure 10 show the variation of the RMSE results of the frequency offset, angle, and time delay of the first path signal with the time delay difference between the two paths. Simulation shows that the method of the present invention is superior to the method of SAGE: taking Δτ=0.25T _s as an example, the accuracy of frequency offset, angle and time delay of the method of SAGE is 2×10 ^-5 /T _s (corresponding to the speed error of 50m/s ), 1.1°, 0.06T _s (corresponding to a distance error of 90cm); while the frequency offset, angle, and time delay accuracy of the method of the present invention are 2.8×10 ^-6 /T _s (corresponding to a speed error of 7m/s), 0.08 °, 4×10 ^-3 T _s (corresponding to a distance error of 6 cm).

references

[1]Indoor Location Market by Component(Technology,Software Tools,andServices),Deployment Mode(Cloud,andOn-premises),Application,Vertical(Transportation,Hospitality,Entertainment,Retail,and Public Buildings),andRegion-Global Forecastto2022[R ], MarketAndMarket, 2020.

[2] D. Chu, "Polyphase codes with good periodic correlation properties (corresp.)," IEEE Transactions on Information Theory, vol.18, no.4, pp.531–532, 1972.

[3] B.H.Fleury, M.Tschudin, R.Heddergott, D.Dahlhaus, and K.Ingeman Pedersen, "Channel parameter estimation in mobile radio environments using the SAGE algorithm," IEEE Journal on Selected Areasin Communications, vol.17, no.3 , pp.434–450, 1999.

[4] S. Boyd and L. Vandenberghe, Convex Optimization. 2004.

Claims (3)

1. A speed, angle and distance joint estimation method based on conjugate ZC sequence pairs is characterized in that a pair of conjugate ZC sequences is designed at a transmitting end to be used as a transmitting sequence; at a receiving end, receiving multipath signals containing different transmission delays, Doppler frequency offsets and angles, estimating parameters by using a maximum likelihood method, and then converting the original high-dimensional parameter estimation problem into a plurality of low-dimensional parameter estimation problems by using an alternative projection method, thereby avoiding high-dimensional search; for single-path parameter estimation, two-dimensional estimation about time delay and frequency offset is converted into two one-dimensional estimation based on the property of a ZC sequence, so that the operation complexity is further reduced, and then accurate value estimation is carried out by combining Newton iteration; the method comprises the following specific steps:

firstly, designing a pair of conjugate ZC sequences;

secondly, based on the property of the conjugate ZC sequence pair, simultaneously obtaining initial solutions of time delay, angle and Doppler frequency offset;

thirdly, performing Newton iteration based on the initial solution of time delay, angle and frequency offset to further obtain the accurate values of three-dimensional parameters of time delay, angle and Doppler frequency offset;

wherein, assuming that a receiving end has a uniform linear array with M antennas, the array response of the signal to the array with respect to the angle θ is expressed as:

where d represents the spacing of the antenna array, λ represents the wavelength, j represents the imaginary number, (. cndot.)^TRepresenting a transpose operation;

considering a multipath environment, a multipath signal received by a receiving end comprises 1 direct path and U-1 reflection paths, each path has different time delay, angle and Doppler frequency offset, and a multipath signal model is as follows:

wherein, beta_u,θ_u,τ_u,ξ_uRepresenting the channel gain, arrival angle, time delay and Doppler frequency offset of the u path signal; y (t) represents a received signal, x (t) represents a training sequence, and z (t) represents gaussian noise subject to a complex gaussian distribution; u is the number of multipath signals;

representing a real number domain;

in the second step, the initial solutions of the time delay, the angle and the Doppler frequency offset are obtained, and the specific process is as follows:

(1) based on model equation (2), considering the first half of the ZC sequence first, the samples of the received signal are expressed as:

wherein,

is an arbitrary real number greater than zero, T_sIs the nyquist sampling period; let T_s1, obtaining:

rewriting (14) to the form:

Y＝A(θ)diag(β)X(τ,ξ)^T+Z (15)

wherein,

X(τ,ξ)＝[x(τ₁)⊙d(ξ₁),x(τ₂)⊙d(ξ₂),…,x(τ_U)⊙d(ξ_U)] (19)

wherein,

representing a complex field, diag (·) represents vector diagonalization as a matrix operation, and an indicates a Hadamard product;

the noise follows a complex gaussian distribution, and the maximum likelihood estimate of the parameters β, τ, ξ, θ can be written in the form of least squares:

wherein τ ═ τ [ τ ]₁,τ₂,…,τ_U]^T,ξ＝[ξ₁,ξ₂,…,ξ_U]^T,θ＝[θ₁,θ₂,…,θ_U]^T,‖·‖_FRepresents the F norm;

because of the fact that

Where vec (-) represents matrix column vectorization,

representing the kronecker product, so there are:

wherein,

therefore, (22) is rewritten as:

wherein |²Which represents the 2-norm of the vector,

the maximum likelihood estimate of β is then expressed as:

wherein, (.)^HRepresenting a transposed conjugation operation, substituting (28) into (26) yields:

wherein,

after estimating the Doppler frequency offset and time delay, according to the velocity

Where c is the speed of light, f_cIs the carrier frequency, thereby estimating the speed of the transmitting end relative to the receiving end according to the distance

The distance can be estimated;

(2) considering a conjugate ZC sequence pair, then:

wherein,

to represent

The projection matrix of (2);

here, x₁(τ_u,∫_u) And

corresponding to the first half ZC sequence, x₂(τ_u,ξ_u) And

corresponding to the second half ZC sequence;

order:

and is

Is to be

From

Deleting the result obtained; using the properties of the projection matrix, we obtain:

wherein,

representation matrix

The orthogonal projection matrix of (a);

substituting equation (37) into equation (29) yields:

assume the multipath number U and

is known, i.e. given

Then equation (38) is reduced to:

wherein, | · | represents modulo; therefore, the multi-path problem is converted into a plurality of single-path problems by using an alternative projection method;

(3) for the solution of the u-th path problem, firstly, estimating initial values of time delay, angle and frequency offset; the initial value of the parameter for the u-th path is achieved using the following approximation:

wherein,

due to the fact that

Then there are:

wherein,

and

are respectively

And

a matrix reconstructed by columns; substituting (41) and (42) into (40) to obtain:

substituting (32) and (33) into (43) to obtain:

order:

then, will be for τ_u,ξ_u,θ_uIs converted into pair eta_u,ζ_u,θ_uEstimation of (2):

first, considering the first half ZC sequence, obtain information about zeta_uAnd theta_uSub-optimal solution of:

by pairs

Two-dimensional fast Fourier transform is carried out, and then the coordinate which enables the module value to be maximum is found out, thereby solving

Consider the second half of the ZC sequence again, at this point

Known, there are:

at this time pair

One-dimensional fast Fourier transform is carried out to find out the coordinate which enables the module value to be maximum, thereby obtaining

Combining (47) and (48) the estimation results to obtain

Further obtaining initial values of time delay and Doppler frequency offset:

。

2. the joint estimation method according to claim 1, wherein the designing of a pair of conjugate ZC sequences in the first step is performed as follows:

(1) for a length of

The ZC sequence of (1):

wherein,

is a positive integer, r is and

a co-prime positive integer parameter;

i.e., the ZC sequence is periodic, the index range of the ZC sequence can be changed: suppose that

Is even, then for an integer delay τ there is:

this means that for a ZC sequence, an integer time delay τ corresponds

The frequency offset of (1); for the

The length is odd, and the time delay-frequency offset interconversion property of the ZC sequence still holds; without loss of generality, let r equal to 1, i.e. r can be omitted;

(2) considering the influence of the shaping filter existing at the transmitting and receiving ends, assuming that the shaping filter existing at the transmitting and receiving ends is a raised cosine filter, the impulse response of the raised cosine filter is expressed as:

where α is the roll-off coefficient, T_sIs the nyquist sampling period; the discrete ZC sequence s (n) is represented as a continuous time signal x (t) after passing through a shaping filter:

due to the low-pass characteristic of the shaping filter, the high-frequency part of the ZC sequence is suppressed; the low frequency part of length L in the middle of the continuous-time signal x (t) is approximated as a chirp signal, i.e.:

wherein L is a positive integer and

the roll-off coefficient α also affects the selection of L;

therefore, the signal of the low frequency part of the ZC sequence passing through the raised cosine filter is regarded as a chirp signal, and there is also an interchange relationship between the time delay and the frequency offset, that is:

the time delay tau can be any and is not limited to be an integral multiple of Nyquist sampling period;

(3) based on the approximation of equation (7), consider a pair of conjugated ZC sequences, denoted s (n) and s, respectively^*(n)：

Wherein,

is an even number and is provided with a plurality of groups,

(·)^*representing a conjugate taking operation; for the first half ZC sequence s (n), a prefix and a suffix with the total length of Q are added when a transmitting end transmits the sequence, and the prefix length is

And the suffix length is

Wherein Q is a positive integer; similarly, the second half of the ZC sequence is also added with a corresponding prefix and suffix; when the receiving end processes, part of the prefix and suffix are removed from the received signal.

3. The joint estimation method according to claim 1, characterized in that in the third step, the accurate values of the three-dimensional parameters are further obtained through newton iteration, as follows:

based on the obtained initial values

Obtaining an accurate value by utilizing Newton iteration; let Λ denote the objective function (39), i.e.:

wherein ψ ═ τ_u,ξ_u,θ_u]^T(ii) a The iteration of newton iterations is:

ψ⁽ⁱ⁺¹⁾＝ψ⁽ⁱ⁾-sH^-1g (51)

wherein,

and

respectively representing a blackplug matrix and a Jacobian vector related to a target function Lambda, and s represents a step length and is obtained by a backtracking straight line method optimization;

the algorithm flow of the alternative projection method is as follows:

when the alternate projection is initialized, assuming that the received signal is single-path, in combination with equation (29), the following equation is used to obtain the three-dimensional parameters:

wherein,

is constant, omitted; thereby estimating out

Wherein the superscript (. circle.) represents the number of iterations, thus, obtaining

Order to

Estimated using equation (39)

Then, let

Thereby estimating

This process is continued until

Estimate out

At iteration 2, first use is made of

Estimated according to equation (39)

Then use

Estimate out

By analogy to utilize

Estimate out

The above process is repeated until the iterations converge.

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